I'd look at making a 3 sided box with the MDF and battons, this wil
also give the table more strength and less bend.

I did think about this, but for a fence, I was just going to clamp one to
the worktop so would want to be able to use G clamps going under the table.
But you did say 3 sided, maybe that'd work.

Alternatively, I could add some 45degree braces both directions on the legs.

I glue everything, for the few pence glue costs why not ?
Yes, unlikely i'd ever need to open it up again.